How can I merge these two spherical objects into a single smooth mesh?
To simple ways, to help keep track of the UV mesh:


In edit mode in the 3D view press Shift+L to select an island of linked vertices. Then hide all other parts with Shift+H. Only the previously selected connected vertices are shown in both 3D and image editor.
When trying to identify meshparts in the uv / image editor, this button may help. 


If your game engine is struggling, deleting additional vertices, which aren't shown anyways, can increase performance. (This also depends on the game engine.) Otherwise, leaving the mesh in whatever state is best maintainable. If joining them is the last step before exporting, I would suggest keeping your work blend file intact and doing mesh joining step separately.

If your performance is fine, hidden faces are no problem.

If you wanted to merge two spheres into a single object and remove the hidden vertices, do the following.


  
  Add a Boolean Modifier to one of the spheres.
  Set the other sphere under the Object in the Modifier Tab.
  Set the Operation to Union
  Apply the modifier.
  
  
  
  
  
  Delete the other sphere.
  


This will cause problem with smooth shading. (Triangles)



With spheres there is an easy fix. We can use the topological features of the spheres to get rid of topology problems. Do the following steps before the boolean operation.


  
  Select one of the spheres and add a Damped Track Modifier, with the other sphere as the Target.
  Do the same vice versa (with the other sphere).
  Set one of the Damped Track Contraints to To: Z and the other to To: -Z.
  
  
  
  
  
  Select both spheres and Apply Visual Transform (Ctrl+A > V). (Optional)
  
  
  
  
  
  You can now delete the constraints.
  
  
  Do the Boolean operation and delete the unused sphere as explained above.
  
  
  In edit mode, select all A then remove doubles W > R
  
  
  


Smooth shading now looks fine, add edge loops to sharpen the crease.



With different objects, you will have to use different approaches to get clean topology with boolean operations.